{
  "id": "1011.3390",
  "version": "v2",
  "published": "2010-11-15T14:16:29.000Z",
  "updated": "2011-03-11T09:39:35.000Z",
  "title": "On the finiteness of the Morse Index for Schrödinger operators",
  "authors": [
    "Baptiste Devyver"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "Let H=$\\Delta +V$ be a Schr\\\"odinger on a complete non-compact manifold. It is known since the work of Fischer-Colbrie and Schoen that the finiteness of the negative spectrum of $H$ implies the existence of a function $\\phi$ solution of $H\\phi=0$ outside a compact set. This has consequences for minimal surfaces and for the finiteness of spaces of harmonic sections in the Bochner method. Here we show that the converse statement also holds: if there exists $\\phi$ solution of $H\\phi=0$ outside a compact set, then $H$ has a finite number of negative eigenvalues.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-11T09:39:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "morse index",
      "schrödinger operators",
      "finiteness",
      "compact set",
      "complete non-compact manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.3390D"
    }
  }
}